This invention relates to Analog to Digital (A/D) conversion.
A/D conversion involves converting a time-continuous and amplitude-continuous signal into a sequence of time-discrete and amplitude-discrete samples and encoding those samples. Conventionally, the signal is sampled in time with a sampling clock having frequency 1/.tau., and uniformly quantized in amplitude with step size q. The accuracy of the conversion depends, of course, both on the resolution in time and the resolution in amplitude. For most practical applications, the signal to be converted is band-limited, and the sampling frequency is at or above the Nyquist rate. Therefore, the time discretization is reversible. The amplitude discretization, however, introduces an irreversible loss of information. This is typically referred to as quantization error.
The prior art has shown that the expected quantization error is ##EQU1##
where s(t) is the time and amplitude-continuous input signal, S.sub.T (t) is a time and amplitude-continuous signal developed from the time discrete and amplitude discrete signal, .tau. is the actual sampling interval, and .tau..sub.N is the Nyquist sampling interval.
The ##EQU2##
quotient in equation (1) suggests that the conversion accuracy can be improved beyond the precision of the quantizer by introducing oversampling. Because the costs involved in building high-precision amplitude quantizers are relatively high, modem techniques for high accuracy A/D conversion are based on oversampling. Apart from this economical and technological consideration, however, using oversampling to improve the conversion accuracy is dramatically inferior, in rate-distortion sense, to refining quantization. For a given precision of the quantizer, unless some entropy coding is used, the bit-rate of oversampled A/D conversion is inversely proportional to the sampling interval, R=O(1/.tau.); hence, the mean squared error decays only inversely to the bit-rate, E(.vertline.s(t)-S.sub.T (t).vertline..sup.2)=O(1/R).
Only recently has it been demonstrated that the accuracy of oversampled A/D conversion is better than suggested by equation (1), and that even without entropy encoding the quantized samples can be efficiently represented so that an exponentially decaying rate-distortion characteristic can be attained. A deterministic analysis shows that if the quantization threshold crossings of a signal s(t) form a sequence of stable sampling for the space of square intergrable bandlimited signals to which s(t) belongs, and s.sub.T (t) is a square intergrable function, in that same bandlimited space, for which the oversampled A/D conversion produces the same digital sequence as for s(t), then the squared error between s(t) and s.sub.r (t) can be bounded as ##EQU3##
uniformly in time. Moreover, representing the information about the quantization threshold crossings requires only a logarithmic increase of the bit-rate as the sampling interval tends to zero. Hence using this quantization threshold crossings based representation the rate distortion characteristic of oversampled A/D conversion becomes EQU .vertline.S(t)-S.sub.T (t).vertline..sup.2 &lt;a.sub.s e.sup.-aR (3)
Where .alpha. is a positive constant, and .alpha..sub.s is proportional to the factor c.sub.s in equation (2).
While these results provide a radically new perspective on oversampled A/D conversion, some important issues remained unanswered. The .tau..sup.2 conversion accuracy is established under the assumption that the quantization threshold crossings of the input signal constitute a sequence of stable sampling in an appropriate class of bandlimited signals. Although there are bandlimited signals which have this property, there also exist bandlimited signals for which the threshold crossings are too sparse to ensure this conversion accuracy; giving a precise characterization of these two classes of bandlimited signals is very intricate. Another problem is that the stronger error bound in equation (2) is not uniform on sufficiently general compact sets for simple oversampled A/D conversion in its standard form. Note also that the result about O(.tau.) accuracy with the linear reconstruction is valid only in a small range of discretization parameters q and .tau., and that the error does not tend to zero along with the sampling interval but rather reaches a floor level for some finite .tau.. So, there are basically no sufficiently general results about the accuracy of simple oversampled A/D conversion. Moreover, no explicit algorithms for reconstructing a bandlimited signal with .tau..sup.2 accuracy are known, and the feasibility of local reconstruction with this accuracy is not clear.